Enumerative combinatorics deals with finite sets and their cardinalities. In other words, a typical problem of enumerative combinatorics is to find the number of ways a certain pattern can be formed.
In the first part of our course we will be dealing with elementary combinatorial objects and notions: permutations, combinations, compositions, Fibonacci and Catalan numbers etc. In the second part of the course we introduce the notion of generating functions and use it to study recurrence relations and partition numbers.
The course is mostly self-contained. However, some acquaintance with basic linear algebra and analysis (including Taylor series expansion) may be very helpful.
Do you have technical problems? Write to us: coursera@hse.ru

We introduce the central notion of our course, the notion of a generating function. We start with studying properties of formal power series and then apply the machinery of generating functions to solving linear recurrence relations.

Taught By

Evgeny Smirnov

Transcript

[SOUND] [MUSIC] Probably, you have already seen power series in your calculus course. They appear as Taylor series of various functions, or in some other ways. And here's a warning. About the difference between formal power series, and a power series from mathematical analysis. That formal power series are actually not functions. So when we write such an expression a naugh + a1q + a2q squared + etc. q is just a formal variable, just a formal letter. We do not assign values to q. We do not compute A(q) for q equal to something. Of course, for some power series, you may do this. Say sometimes, We can treat A(q) as a function. For some values, Of q. And study its convergence. For instance, if you take the geometric progression, 1 + q + q squared + etc., this converges for, If the absolute value of q is Less than 1. But we are not going to do that, we are not studying convergence. Say if, well, of course, you can invent formal power series with coefficients growing so rapidly that this series never converges outside q = 0. So an example is if you take an equal to n to the power n. A(q), Which is sum of n to the power n times q to the power n, does not converge anywhere. Its coefficients grow too rapidly. More rapidly than any power of q. Not converge outside q equal to 0. But this is a formal power series in its own right, but, So you can multiply this formal power series by any other formal power series. Or, Add it with something or do whatever you like. Okay, but sometimes, analytic arguments can be useful. Well, they can give us some hints. How to do, how to deal with formal power series. Okay, here is an example of such a hit. Sometimes, ideas from calculus can help us dealing with a formal power series. Say, [COUGH] consider the following example. Let us take the power series 1 + 2q + 3q squared + 4q to the third + etc. And let us find its inverse. Denote the series by A(q) as usual. Okay, if you look at this expression, you can see that this looks very much like a derivative of the formal power series. So suppose that B(q) = 1 + q + q squared + etc. Suppose that these were an actual function depending on the real parameter q. And defined inside the interval, absolute value of q is less than 1. In this case, we can take its derivative. And well, a theory from mathematical analysis tells us that we can derive such power series component wise. The derivative of this power series is, 1 + 2q + 3q squared + etc. This is exactly A(q). On the other hand, we know that B(q) = 1 divided by 1 over q. And so if we still treat this as a function from negative 1 to 1, this means that A(q) is the derivative of 1 over 1- q. And this can be found by the chain root. So this will be = 1 over 1- q squared. So now, if we check this equality, that the A inverse, A(q) inverse = (1-q) squared. So in this case, a theorem from mathematical analysis allowed us to guess what was the right answer. Well, of course, this can be formalized in the formal power series setting. Of course, you can define the formal derivative of a power series. So if for a formal power series, A(q) a naught + a1q + a2q squared + etc. Its formal derivative, its derivative is defined by, Defined as A'(q). Just component wise, it is defined as a1 + 2a2 times q + 3a3q squared + etc + ananq to the power n- 1 + and so on. So this is a formal definition of the derivative of A(q). And if you have now equalities in formal power series, then you can basically do all the same as with usual functions. So for them, you have the say, you have the chain rule. You have the equality, (A(q) times B(q))' = A(q) times B'(q) + plus A'(q) times B(q), and so on. Well, I leave the proof of this e quality as an exercise for you. And another exercise which I'm going to fit in here is as follows. And another exercise. Find the power series is as follows, find the power series expansion, Of 1 over 1- q to the power of K. Hint. Use the derivative as in this example. [SOUND] [MUSIC]

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